TITLE: Finding Two Rainbow Spanning Trees
QUESTION [10 upvotes]: Suppose we have a graph whose edges are coloured. It's not necessarily a proper colouring: a given node may have 0, 1, or several incident edges of a given colour.
Is the following problem NP-complete? Determine whether there are two edge-disjoint spanning trees, such that in each individual tree, no colour appears twice.
I am curious because the variant "determine whether there are two edge-disjoint spanning trees, such that in the union of the trees, no colour appears twice" is solvable in polynomial time, for example using matroid theory.

REPLY [2 votes]: Here are some nice lemmas that you can use:
http://www.cs.elte.hu/egres/qp/egresqp-10-04.pdf
Dave pointed out my mistake, the subgraphs of spanning trees do not have to be trees. So I have no clue about the answer.
Espacially Problem 4 (or 10) seems promising. Take the graph from their construction such that all of its edges have a different color, suppose it has e edges. Then if we allow multigraphs, adding every edge with multiplicity 2n-2-e, all of a different color but same for each edge (thus in total we have 2n-2 colors) shows that your question solvable in P is NP-hard for multigraphs. Am I right? I would guess that with some further tricks you can make a simple graph from this for the variant you asked.